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**Numerical sequences**with 11th grade math worksheets online to progress in high school math.

Exercise 1 – Solving an equation using sequences

Solve the equation:

Hint: calculate the sum and then notice that if x is a solution then x < 0.

Exercise 2 – Sum of squares

Calculate the following sum:

Hint: Group the terms in pairs.

Exercise 3 – Sum of even and odd integers

Calculate the following sums:

sum of the first odd natural numbers.

sum of the first even natural numbers.

Exercise 4 – Study of a numerical sequence

Let be the sequence defined by :

.

1. Calculate .

2. Is the sequence arithmetic ?

Exercise 5 – Arithmetic or geometric sequence

Consider the sequence defined by .

1. Calculate

2. Is the sequence arithmetic ? Geometric ?

Exercise 6 – Study of two sequences

Consider the two sequences and defined for all by :

.

1. Let be the sequence defined by .

Prove that is a geometric sequence.

Exercise 7 – Geometric sequence, study

Consider the geometric sequence with first term and reason .

1. Calculate

2. Calculate .

3. Calculate the sum .

Exercise 8 – Square roots

Let be the sequence defined for all n by .

1. Using your calculator, calculate .

What conjecture can be made about the direction of variation of the sequence? For a possible limit?

2. Prove that for all n not zero,

.

3. Deduce the direction of variation of the sequence .

4. Using the result of question 2., show that for any non-zero natural number n,

.

5. Deduce that the sequence is convergent and specify its limit.

Exercise 9 – Study of an arithmetic sequence

The sequence is arithmetic of reason .

It is known that and .

1. Calculate the reason and

2. Calculate the sum .

Exercise 10 – Calculating a sum of numbers

Calculate the following sum:

Exercise 11 – Graphical representation of a sequence

Consider the sequence defined for any non-zero natural number by the relation: .

1. Prove that the sequence is arithmetic of reason r which we will specify. Specify its direction of variation.

2. Graphically represent the sequence .

We will limit ourselves to the first five or six terms.

Exercise 12 – Determine a sum of integers

Calculate the exact value of the sum :

Exercise 13 – Book reading

John is reading a book.

By adding up the numbers of all the pages he has already read, he gets 351.

By adding up the numbers of all the pages he read

i remains to be read, it gets 469.

1. What page is John on?

2. How many pages does this book have?

Note: we will assume that the book starts on page 1.

Exercise 14 – Determining a number

Determine a number x such that the three numbers 25, x and 16 are three consecutive terms of a geometric sequence of negative reason.

Exercise 15 – Problem on numerical sequences

A student rents a room for 3 years.

We offer two types of leases:

1st contract: a rent of 200€ for the first month then an increase of 5€ per month until the end of the lease.

2nd contract: a rent of 200 € for the first month then an increase of 2% per month until the end of the lease.

1. Calculate, for each of the two contracts, the second month’s rent and then the third month’s rent.

2. Calculate, for each of the two contracts, the rent for the last month, i.e. the 36th month.

3. What is the most advantageous contract overall for a 3-year lease? Justify with calculations.

*Vocabulary: a lease is a rental agreement.*

Exercise 16 – Rectangle

1. ABC is a right triangle.

Its smallest side is 1 and the lengths of its sides are three terms

of an arithmetic sequence.

Determine these lengths.

2. ABC is a right triangle.

Its smallest side is 1 and the lengths of its sides are three terms

of a geometric sequence.

Determine these lengths.

Exercise 17 – Double recurrence sequence

Consider the sequence defined by recurrence by :

1. Calculate

2. Solve the following second degree equation: .

3. Determine two real numbers A and B such that: .

4. From this, deduce

Exercise 18 – Calculate the limit

Determine the limit of the sequence defined by :

for all .

Exercise 19 – Study of a sequence and demonstration by recurrence

Consider the sequence defined by recurrence by :

1. Calculate

2. Prove by recurrence that for all

Exercise 20 – Determine the value of two numerical expressions

Calculate the exact value of the following numbers:

Exercise 21 – Arithmetic sequence

Consider u(n), an arithmetic sequence of reason r.

1°) Justify that u(3) = u(2) + r and that u(4) = u(3) + r

Deduce that u(4) = u(2) + 2r

2°) Show that u(8) = u(5) + 3r

3°) What relationship can be written between u(7) , u(2) and r ? Justify.

4°) It is assumed in this question that u(0) = 4 and r = 2.

Calculate u(5) .

Give without demonstration the value of u(100).

Exercise 22 – Graphical representation

We define a sequence (un) by: un = 17 243 – 8n for any integer n.

For example, replacing n by 10: u10 = 17,243 – 8 x 10 = 17,163

1°) Calculate u0 ; u1 ; u1990 ; u1991 ; u1992 .

2°) Calculate u1 – u0 ; u1991 – u1990 ; u1992 – u1991

3°) By replacing n by n+1 in the expression of one show that

for any integer n: un+1 = 17 235 – 8n

Deduce that, for any integer n: un+1 – un = -8

4°) Using the relation one+1 – one = -8, that is to say one+1 = one – 8 complete the following table.

Is the sequence (un) a decreasing sequence ?

n |
1990 |
1991 |
1992 |
1993 |
1994 |
1995 |
1996 |
1997 |
1998 |
1999 |
2000 |

a |
1 323 |

5°) Graph the sequence (un) when n varies from 1990 to 2000.

Fiscal Year 23 – List of Electors

The following table shows the number of registered voters in a small community for the years 1990 to 2000.

Year |
1990 |
1991 |
1992 |
1993 |
1994 |
1995 |
1996 |
1997 |
1998 |
1999 |
2000 |

Number of registrants |
1323 |
1313 |
1304 |
1297 |
1288 |
1289 |
1281 |
1271 |
1258 |
1248 |
1243 |

1°) We note Pn the number of registered voters for the year n.

Give the value of P1992 and P1998

2°) Calculate P1994 – P1993. What does this number represent?

Calculate P1995 – P1994. What does this number represent?

3°) Can we say that the sequence of numbers Pn is a decreasing sequence when n varies from 1990 to 2000 ?

4°) Represent graphically the sequence (Pn).

Exercise 24 – Study of a capital

We have a capital of €.

On January 1, 2000, this capital is placed in an account that is compounded at 3% per year.

1. Calculate the capital obtained after one year.

2. Calculate the capital obtained after 7 years.

By what percentage did the capital increase during these 7 years?

3. How many years do you need to leave this money in the account in order to have a capital of at least 2 000 €?

Exercise 25 – Numerical sequences and percentages

Cosmic rays continuously produce carbon-14 in the atmosphere, which is a radioactive element.

During their lifetime, animal and plant tissues contain the same proportion of carbon 14 as the atmosphere.

This proportion decreases after the death of the tissue by 1.14% in 100 years.

1. Determine the percentages of the initial proportion of carbon-14 contained in the tissue after 1,000 years, 2,000 years and 10,000 years.

2. Express the percentage of the initial proportion of carbon-14 contained in the tissue after years.

3. A fossil contains only 10% of what it should contain in carbon 14.

Give an estimate of his age.

Exercise 26 – Problem

“On the first day of the month, I earned 2 cents;

On the second day of the month, I earned 4 cents;

On the third day of the month, I earned 8 cents;

etc … doubling from one day to the next.

At the end of the month, I had earned about a billion cents!

It was towards the end of the sixties … ”

What year was it?

Exercise 27 – Remuneration in a company

A company, offers to recruit a new employee two types of compensation:

Type 1: Initial salary of €1,200 per month with an annual increase in monthly salary of €100.

Type 2: Initial salary of €1,100 per month with an annual salary increase of 8%.

1) In the case of type 1 remuneration, we note u(0) the initial monthly salary, and u(n) the monthly salary after n years. Give the values of u(0), u(1), u(2).

2°) In the case of type 2 remuneration, we note v(0) the initial monthly salary, and v(n) the monthly salary after n years. Give the values of v(0), v(1), v(2).

3°) Give a general expression for u(n) and v(n) as a function of n. Compute u(5) and v(5); u(8) and v(8).

4°) The new employee intends to stay 10 years in the company. What is the most advantageous remuneration?

Exercise 28 – Population of a village

One village had 3123 inhabitants in 1995. The number of inhabitants decreases by 12% every year.

We note P(n) the number of inhabitants of the village for the year n.

1°) Give the values of P(1995) and P(1996). (round up to the nearest integer)

2°) Justify that the sequence P(n) is a geometric sequence and give its reason.

3°) Calculate P(2001). (round up to the nearest integer)

4°) In what year will the number of inhabitants have decreased by two thirds compared to 1995?

5°) Represent graphically the sequence P(n) for n varying from 1995 to 2005.

Exercise 29 – Geometric sequence

Consider v(n) a geometric sequence of reason q.

1°) Justify that v(3) = v(2) x q and that v(4) = v(3) x q

Deduce that v(4) = v(2) x q2

2°) Show that v(8) = v(5) x q3

3°) What relationship can be written between v(7) , v(2) and q ? Justify.

4°) It is assumed in this question that v(0) = 3 and q = 2.

Calculate v(5) .

Give without demonstration the value of v(100) .

Exercise 30 – Capital and numerical series

A capital of 12,618 euros is invested on 01/01/2000 with an annual interest rate of 6.3%.

Every year the interest is added to the capital.

We note C(0) the capital corresponding to January^{1,} 2000. We have therefore C(0) = 12 618.

For any integer n, C(n) is the capital corresponding to January^{1st of} the year 2000+n.

1°) Calculate C(1), C(2), C(3). (results will be rounded to the nearest euro cent)

2°) Prove that for any integer n we have C(n+1) = C(n) x 1.063.

3°) Complete the following table (results should be rounded to the nearest euro cent)

Rank n of the year |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |

Capital C(n) |
12 618 |

4°) Represent graphically the sequence C(n).

Exercise 31 – Calculating the first term of an arithmetic sequence

Let be an arithmetic sequence of reason and such that .

Calculate the value of the first term .

Exercise 32 – A recurring sequence that is arithmetic

Consider the sequence defined by .

1. Calculate

2. Justify that the sequence is an arithmetic sequence whose reason is specified.

3. What is worth?

Exercise 33 – Calculating a sum

Consider the sequence defined by .

1. Calculate

2. Is the sequence arithmetic ? If yes, specify the reason.

3. What is worth?

4. Calculate the sum .

Exercise 34 – Arithmetic sequence and sum of terms

Consider defined by .

1. Calculate

2. Prove that is an arithmetic sequence whose reason is specified.

3. What is worth?

Calculate .

Exercise 35 – Arithmetic sequences and problem

The triodule is a weed: it produces a single seed during its first year of growth which it sends far enough away (this one will germinate at the beginning of the following year) and it develops to occupy the surface of 1m².

In the following years, the foot is content to increase its surface by 1m².

The first and only triodule seed arrived in 1800 and germinated in the spring of 1801 on Blécarre Island.

Questions:

1/ a) What area will the triodule foot occupy at the end of the year?

b) What will happen in 1802?

c) What area will the old triodule foot occupy at the end of 1802?

2/ Prepare a spreadsheet:

In C2, write: =B2+1 ,then using the copy handle, complete the cells in row number 2.

In A4, write: =A3+1.

In B4, write: =B3+1.

In C4, write: =B3 ,then copy this formula from 40 cells to the right.

Finally, copy line 4 down.

3/ Let An be the area occupied by all the triodule feet at the end of the year 1800+n. It is assumed that each seed produced has developed a foot.

a) Give the value of A0, A1 and A2 by inserting a new column in the spreadsheet.

(b) What is the area of the first triodule foot at the end of the year 1800+n?

c) Verify that we have An=1+2+3+….+n.

d) Using the worksheet, give the area occupied by all the triodule feet after 20 years.

e) In what year will the total triodule footprint exceed 500m²?

Exercise 36 – Study of the nature of a sequence

Study the nature of the following sequences:

a) for any natural number n, .

b) for any natural number n, .

Exercise 37 – Numerical sequences

We note the sequence defined by :

1. Calculate

2. Express as a function of .

3. Express as a function of .

4. Deduce the expression of , as a function of

(We will not demonstrate the equality found).

5. Calculate .

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